Calculation of the boundary layer on the electrically conducting wall of a plane channel

There are presently available quite a large number of works devoted to the study of the motion of an electrically conducting fluid in boundary layers formed on electrodes or on the nonconducting walls of various MHD devices. However, the methods of solving the boundary layer equations in these studies are based on various simplifying assumptions which allow the problem to be reduced to the solution of a system of ordinary differential equations. Thus, in [1] there is imposed on the flow the special magnetic fieldH1/x, which enables the problem to be reduced to the self-similar form, while in the studies of other authors [2, 3] either the solution is sought in the form of expansions in x, or it is assumed that the problem is locally self-similar [4]. In the present paper we construct the solution of the MHD boundary layer equations which is obtained by one of the numerical methods which has long been used for solving the boundary layer equations for a nonconducting fluid.     

The establishment of boundary integral equations by generalized functions

By the theory of generalized functions this paper introduces a specific generalized function _p by which, together with its various derivatives, the boundary integral equations and its arbitrary derivatives of any sufficiently smooth function can be established. These equations have no non-integral singularities. For a problem defined by linear partial differential operators, the partial differential equations of the problem can always be converted into boundary integral equations so long as the relevant fundamental solutions exist.This paper is partial fulfilment of the first author's doctorial dissertation and the second author is the endviser.     

Memory effects in a non-uniform flow: A study of the behaviour of a tubular film of viscoelastic fluid

Summary Formal use of constitutive equations such as that ofOldroyd in the mathematical model of a flow leads, in general, to a higher order differential equation than is obtained for a purely viscous fluid, and so we expect to need more boundary conditions in order to specify the problem completely. (These extra boundary conditions may be thought of as arising from the need to specify what the fluid remembers of the flow outside the region of interest.) In flows which are uniform spatially, or uniform with time for a material element, the uniformity will provide the extra information and so no extra conditions are needed. Similarly for confined flows, where no new fluid enters the region of interest, no information about flow outside this region is needed.Here the steady flow of a tubular film of a viscoelastic fluid is studied with the particular aim of examining the effect of these extra boundary conditions in a situation where they may be expected to have some significant influence on the flow as a whole. The flow, while being geometrically complex, is essentially an elongational free-surface flow involving the biaxial stretching of a thin axisymmetric tubular film. Features of the constitutive equations studied are the presence of a non-zero relaxation time and the possibility of a variable viscosity. One effect of the non-zero relaxation time is that a tube of constant radius (possible but unstable for aNewtonian fluid) is not dynamically possible. Preliminary computational results suggest that the effect of the extra upstream boundary conditions is not large, and also have failed to show any major difference between the two generalisations of theMaxwell model which have been used.With 1 figure     

Use of turbulent energy balance equation in the theory of turbulent flows along walls

The turbulent flow of an incompressible fluid is considered in a plane channel, a circular tube, and the boundary layer on a flat plate. The system of equations describing the motion of the fluid consists of the Reynolds equations and the mean kinetic energy balance equation for turbulent fluctuations. On the basis of an analysis of experimental data, hypotheses are formulated with respect to the eddy kinematic viscosity and lengthl entering into the expression for specific dissipation of turbulent energy into heat. It is assumed that in the central (outer) region of the flow in a channel, andl are constants, and expressions are taken for them which are used for a free boundary layer; near the walll varies linearly and almost linearly. Results of calculations of the turbulent energy distribution, the mean velocity, and the drag coefficient are in good agreement with the existing experimental data. The values of two empirical coefficients, which enter into the system of equations as the result of the hypotheses, are close to those obtained for a free boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–33, May–June, 1973.     

Displacement type unified equation for bending, buckling and vibration of conical shells and its applications

By using Donnell's simplication and starting from the displacement type equations of conical shells, and introducing a displacement functionU(s,,) (In the limit case, it will be reduced to cylindrical shell displacement function introduced by V. S. Vlasov) and a generalized loadq,(s,,),the equations of conical shells are changed into an eighth—order solvable partial differential equation about the displacement functionU(s,,). As a special case, the general bending problem of conical shells on Winkler foundation has been studied. Detailed numerical results and boundary coefficients for edge unit loads are obtained.The project supported by the National Natural Science Foundation of China.     

Two-component modeling for non-Newtonian nanofluid slip flow and heat transfer over sheet: Lie group approach

The present paper deals with the multiple solutions and their stability analysis of non-Newtonian micropolar nanofluid slip flow past a shrinking sheet in the presence of a passively controlled nanoparticle boundary condition. The Lie group transformation is used to find the similarity transformations which transform the governing transport equations to a system of coupled ordinary differential equations with boundary conditions. These coupled set of ordinary differential equation is then solved using the RungeKutta-Fehlberg fourth-fifth order(RKF45) method and the ode15 s solver in MATLAB.For stability analysis, the eigenvalue problem is solved to check the physically realizable solution. The upper branch is found to be stable, whereas the lower branch is unstable. The critical values(turning points) for suction(0 sc s) and the shrinking parameter(χc χ 0) are also shown graphically for both no-slip and multiple-slip conditions. Multiple regression analysis for the stable solution is carried out to investigate the impact of various pertinent parameters on heat transfer rates. The Nusselt number is found to be a decreasing function of the thermophoresis and Brownian motion parameters.     

Calculation of viscous compressible gas flow past a corner

We consider the problem of calculating the parameters for supersonic viscous compressible gas flow past a corner (angle greater than ). The complete system of Navier-Stokes equations for the viscous compressible gas is solved in the small vicinity Q₁. (characteristic dimensionl~1/R) of the corner point. The conditions for smooth matching of the solution of the Navier-Stokes equations and the solution of the ideal gas or boundary layer equations are specified on the boundary of Q₁. All these solutions are a priori unknown, and the conditions for smooth matching reduce to certain differential equations on the boundary of Q₁. Here account is taken of the interaction of the flows near the wall surface and in the so-called outer region [1].We note that no a priori assumptions are made in Q₁ concerning the qualitative behavior of the solution, in contrast with other studies on viscous flow past a corner (for example, [2–4]).The Navier-Stokes system in Q₁ is solved numerically, using the difference scheme suggested in [5]. This scheme permits obtaining the steady-state solution by the asymptotic method for large Reynolds numbers R, and also has an approximation accuracy adequate to account for the effects of low viscosity and thermal conductivity.     

Gaussian Closure of One-Dimensional Unsaturated Flow in Randomly Heterogeneous Soils

We propose a new method for the solution of stochastic unsaturated flow problems in randomly heterogeneous soils which avoids linearizing the governing flow equations or the soil constitutive relations, and places no theoretical limit on the variance of constitutive parameters. The proposed method applies to a broad class of soils with flow properties that scale according to a linearly separable model provided the dimensionless pressure head has a near-Gaussian distribution. Upon treating as a multivariate Gaussian function, we obtain a closed system of coupled nonlinear differential equations for the first and second moments of pressure head. We apply this Gaussian closure to steady-state unsaturated flow through a randomly stratified soil with hydraulic conductivity that varies exponentially with where =(1/) is dimensional pressure head and is a random field with given statistical properties. In one-dimensional media, we obtain good agreement between Gaussian closure and Monte Carlo results for the mean and variance of over a wide range of parameters provided that the spatial variability of is small. We then provide an outline of how the technique can be extended to two- and three-dimensional flow domains. Our solution provides considerable insight into the analytical behavior of the stochastic flow problem.     

Non-Newtonian fluids v frictional resistance of discs and cones rotating in power-law non-Newtonian fluids

Summary Relations have been derived for the frictional resistance of finite discs and cones rotating in Ostwald-de Waele (power-law) type non-Newtonian fluids. The obtained equations can be formulated as dimensionless relations between the dimensionless moment coefficient and the generalized Reynolds number; the flow-behaviour index n enters the equations as a parameter. The relations derived for cones contain the apex angle 2₀ as an additional parameter in the form of A=sin ₀. The validity of the theoretically derived relations has been verified by measurements of the torque of discs and cones for a number of pseudoplastic power-law fluids.Nomenclature A sin ₀ parameter - b exponent in regression equation (16) - C coefficient in regression equation (16) - c _Mi dimensionless moment coefficient, for bodies wetted on one side (i=1) and for completely wetted bodies (i=2), equations (8) and (9b) - d diameter of turntable - F, G velocity functions of exact solution, equation (4) - K consistency coefficient of non-Newtonian fluids - M _Ki torque of rotating bodies, i=1 for bodies wetted on one side, i=2 for completely wetted bodies - n flow-behaviour index of non-Newtonian fluids - N=K/ kinematic consistency coefficient - P tangential force - r(y) perpendicular distance of point on cone surface from axis - R radius of disc or of base of cone - modified Reynolds number defined by equation (14) - Re _ow generalized Reynolds number defined by equation (10) - S, S area - u, v components of velocity vector - x, y, z coordinates according to fig. 1 - ₀ half the apex angle of cone - coefficient of frictional resistance defined by equation (11) - thickness of boundary layer - independent variable in exact solution, defined by equation (5) - density of fluid - _zx, _zy tangential stresses - angular velocity of rotation Indices T theoretical value - E experimental value - 0 refers to surface of rotating body     

The Stress–Strain State of Geometrically and Physically Nonlinear Plates with One-Sided Corrosive Wear

A technique is proposed to investigate one-sided corrosive wear. The problem is solved with regard for geometric and physical nonlinearity. Two, Dolinskii's and Gutman's corrosion models are considered. The quasistatic problem is solved by the method of variational iterations, which reduce ordinary differential equations to a system of nonlinear equations with approximation o(h ²) to be solved by Newton's method. At each step, to allow for physical nonlinearity, the method of variable elastic parameters is used. Also a technique is developed to consider various boundary conditions and ⁱ(e ⁱ) diagrams. Specific numerical results are presented.     

Method of solution of the hypersonic boundary layer equations in the case of strong viscous-inviscid interaction

A method of solving the boundary layer equations is developed taking into account the strong interaction between the boundary layer and the outer hypersonic inviscid flow. The method is aimed at solving problems whose salient feature is the possible upstream propagation of disturbances over distances comparable with the body length. The procedure for fitting a self-consistent contour of the effective body using an artificially formulated boundary value problem for an ordinary second-order differential equation, which lies at the basis of the method, is considered in detail. The method is applied to the problem of flow around a flat plate with roughness in the form of an embankment or a trench; the calculated results are presented.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 81–89, July–August, 1995.     

A remark on similarity analysis of boundary layer equations of a class of non-Newtonian fluids

A similarity analysis of three-dimensional boundary layer equations of a class of non-Newtonian fluid in which the stress, an arbitrary function of rates of strain, is studied. It is shown that under any group of transformation, for an arbitrary stress function, not all non-Newtonian fluids possess a similarity solution for the flow past a wedge inclined at arbitrary angle except Ostwald-de-Waele power-law fluid. Further it is observed, for non-Newtonian fluids of any model only $90°$ of wedge flow leads to similarity solutions. Our results contain a correction to some flaws in Pakdemirli׳s [14] (1994) paper on similarity analysis of boundary layer equations of a class of non-Newtonian fluids.     

Squeeze flow of soft solids between rough surfaces

Newtonian liquids and non-Newtonian soft solids were squeezed between parallel glass plates by a constant force F applied at time t=0. The plate separation h(t) and the squeeze-rate were measured for different amplitudes of plate roughness in the range 0.3–31 m. Newtonian liquids obeyed the relation Vh ³ of Stephan (1874) for large plate separations. Departures from this relation that occurred when h approached the roughness amplitude were attributed to radial liquid permeation through the rough region. Most non-Newtonian materials showed boundary-slip that varied with roughness amplitude. Some showed slip that varied strongly during the squeezing process. Perfect slip (zero boundary shear stress) was not approached by any material, even when squeezed by optically-polished plates. If the plates had sufficient roughness amplitude (e.g. about 30 m), boundary slip was practically absent, and the dependence of V on h was close to that predicted by no-slip theory of a Herschel-Bulkley fluid in squeeze flow (Covey and Stanmore 1981, Adams et al. 1994).     

Stability of subsonic gasdynamic flows

A system of differential equations describing small perturbations of the steady flow of a non-viscous ideal gas in a channel of variable cross section is analyzed in this paper. The equations of nonsteady flow and the boundary conditions are linearized, and the solution of the linearized equations is sought in the form v(x)expt t, where v(x) is an eigenfunction while is the natural frequency for the boundary problem being studied. With such an approach the problem is reduced to finding the solutions to ordinary differential equations with variable coefficients which depend on the parameter . Analytical solutions of this system are obtained for small values of and for values of ¦¦1. The results can be used to calculate the growth of high-frequency and low-frequency perturbations imposed on subsonic, supersonic, and mixed (i.e., with transitions through the velocity of sound) gasdynamic flows, to analyze the stability of subsonic sections, and to verify and supplement various numerical methods for calculating unsteady flows and numerical methods for studying stability in gasdynamics. The application of the solutions found for small and large is demonstrated on a study of flow stability behind a shock wave (a direct compression shock in the present formulation). Analytical expressions are obtained for the determination of from which it follows that the flow stability behind a shock essentially depends on the shape of the channel at the place where the shock is located in the steady flow, which was noted earlier in [1], and on the conditions of the reflection of small perturbations in the exit cross section of the channel, which was first pointed out in [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 90–97, January–February, 1978.In conclusion, the author thanks A. G. Kulikovskii and A. N. Sekundov for helpful discussions of the work.     

p-version least squares finite element formulation for two-dimensional,incompressible, non-Newtonian isothermal and non-isothermal fluid flow

This paper presents a p- version least squares finite element formulation (LSFEF) for two-dimensional, incompressible, non-Newtonian fluid flow under isothermal and non-isothermal conditions. The dimensionless forms of the diffential equations describing the fluid motion and heat transfer are cast into a set of first-order differential equations using non-Newtonian stresses and heat fluxes as auxiliary variables. The velocities, pressure and temperature as well as the stresses and heat fluxes are interpolated using equal-order, C⁰-continuous, p-version hierarchical approximation functions. The application of least squares minimization to the set of coupled first-order non-linear partial differential equations results in finding a solution vector {δ} which makes the partial derivatives of the error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. The paper presents the implementation of a power-law model for the non-Newtonian Viscosity. For the non-isothermal case the fluid properties are considered to be a function of temperature. Three numerical examples (fully developed flow between parallel plates, symmetric sudden expansion and lid-driven cavity) are presented for isothermal power-law fluid flow. The Couette shear flow problem and the 4:1 symmetric sudden expansion are used to present numerical results for non-isothermal power-law fluid flow. The numerical examples demonstrate the convergence characteristics and accuracy of the formulation.     

Darcy-Forchheimer natural,forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.Nomenclature A cross-sectional area - b _i coefficient in the chosen temperature profile - B ₁ coefficient in the profile for the dimensionless boundary layer thickness - C coefficient in the modified Forchheimer term for power-law fluids - C ₁ coefficient in the Oseen approximation which depends essentially on pore geometry - C _i coefficient depending essentially on pore geometry - C _D drag coefficient - C _t coefficient in the expression forK ^* - d particle diameter (for irregular shaped particles, it is characteristic length for average-size particle) - f _p resistance or drag on a single particle - F _R total resistance to flow offered byN particles in the porous media - g acceleration due to gravity - g _x component of the acceleration due to gravity in thex-direction - Grashof number based on permeability for power-law fluids - K intrinsic permeability of the porous media - K ^* modified permeability of the porous media for flow of power-law fluids - l _c characteristic length - m exponent in the gravity field - n power-law index of the inelastic non-Newtonian fluid - N total number of particles - Nux,_D,F local Nusselt number for Darcy forced convection flow - Nux,_D-F,F local Nusselt number for Darcy-Forchheimer forced convection flow - Nux,_D,M local Nusselt number for Darcy mixed convection flow - Nux,_D-F,M local Nusselt number for Darcy-Forchheimer mixed convection flow - Nux,_D,N local Nusselt number for Darcy natural convection flow - Nux,_D-F,N local Nusselt number for Darcy-Forchheimer natural convection flow - pressure - p exponent in the wall temperature variation - _Pe c characteristic Péclet number - _Pe x local Péclet number for forced convection flow - _Pe x modified local Péclet number for mixed convection flow - _Ra c characteristic Rayleigh number - _Ra x local Rayleigh number for Darcy natural convection flow - _Ra x local Rayleigh number for Darcy-Forchheimer natural convection flow - Re convectional Reynolds number for power-law fluids - Reynolds number based on permeability for power-law fluids - T temperature - T _e ambient constant temperature - T w,_ref constant reference wall surface temperature - T _w(X) variable wall surface temperature - T _w temperature difference equal toT w,_ref–T _e - T ₁ term in the Darcy-Forchheimer natural convection regime for Newtonian fluids - T ₂ term in the Darcy-Forchheimer natural convection regime for non-Newtonian fluids (first approximation) - T _N term in the Darcy/Forchheimer natural convection regime for non-Newtonian fluids (second approximation) - u Darcian or superficial velocity - u ₁ dimensionless velocity profile - u _e external forced convection flow velocity - u _s seepage velocity (local average velocity of flow around the particle) - u _w wall slip velocity - U c _M characteristic velocity for mixed convection - U c _N characteristic velocity for natural convection - x, y boundary-layer coordinates - x ₁,y ₁ dimensionless boundary layer coordinates - X coefficient which is a function of flow behaviour indexn for power-law fluids - effective thermal diffusivity of the porous medium - shape factor which takes a value of/4 for spheres - shape factor which takes a value of/6 for spheres - ₀ expansion coefficient of the fluid - _T boundary-layer thickness - T ₁ dimensionless boundary layer thickness - porosity of the medium - similarity variable - dimensionless temperature difference - coefficient which is a function of the geometry of the porous media (it takes a value of 3 for a single sphere in an infinite fluid) - ₀ viscosity of Newtonian fluid - ^* fluid consistency of the inelastic non-Newtonian power-law fluid - constant equal toX(2 ^2–n )/ - density of the fluid - dimensionless wall temperature difference     

Approximate solution of the boundary problem for the equations of a stationary electrostatic charged-particle beam

The solution is given of the equations of a three-dimensional stationary electrostatic beam of charged particles of like sign filling the region between two nearby curvilinear surfaces. We assume that the flow is nonrotational and nonrelativistic and that the velocity vector is a single-valued function. The solution is constructed in the form of an asymptotic series in powers of the small parameter , which is the ratio of the characteristic transverse (a) and longitudinal (l) dimensions of the problem. The first dimension is taken to be the distance between the electrodes, andl defines the scale at which the geometric and physical parameters (emitter curvature, electric field E on the emitter, and the emission current density J) change noticeably. The emission regimes limited by the space charge (-regime), temperature (T-regime), and the case of nonzero initial velocity (U-regime) are studied. The asymptotic behavior is given by the formulas for the corresponding one-dimensional flow between parallel surface.The solution of the boundary problem for emission in the-regime reduces to determination of the emission current density J for fixed electrode geometry and given accelerating voltage. The corresponding formulas are presented, retaining terms of order ³.Two approximations with respect to are performed for the T- and U-regimes. Here the unknown quantity for given properties of the emitting surface (J) will be the electric field E.The results provided by the constructed expansions are compared with the exact solution for flow from a planar emitter along circular trajectories [1]. As an example we examine the two-dimensional problem of flow between two nearby circular cylindrical electrodes with disruption of the coaxiality.The conventional tensor notations are used.     

Numerical investigations of flow of a viscoelastic fluid between rotating coaxial disks

Summary A numerical investigation is made of the problem of flow of a viscoelastic fluid of the Rivlin-Ericksen type between rotating coaxial disks. The finite-difference analogues of the governing nonlinear ordinary differential equations are written and the resulting equations are solved using point successive overrelaxation method (SOR) under the appropriate boundary conditions. Two cases of interest are treated, namely, when one of the disk is at rest while the other rotates with a constant angular velocity and when both the disks rotate with constant angular velocities but in the opposite directions. Typical examples of Reynolds numberR in the range 10 R 1000 are described for various values of the non-Newtonian parameters and results are compared with those for a classical viscous fluid.

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Zusammenfassung Die Strömung einer viskoelastischen Flüssigkeit vom Rivlin-Ericksen-Typ zwischen zwei rotierenden koaxialen Kreisscheiben wird numerisch untersucht. Die das Problem beschreibenden nichtlinearen Differentialgleichungen werden in analogen Gleichungen für finite Differenzen umgeschrieben und unter angepaßten Randbedingungen mit einem punktweisen Über-Relaxations-Verfahren (SOR) gelöst. Es werden zwei interessierende Fälle behandelt, zuerst der Fall, bei dem eine Platte ruht und die andere mit konstanter Winkelgeschwindigkeit rotiert, als zweiter derjenige, bei dem beide Platten mit gleichen konstanten Winkelgeschwindigkeiten, jedoch entgegengesetztem Drehsinn rotieren. Typische Beispiele für Reynoldszahlen zwischen 10 und 1000 werden für verschiedene Werte der nicht-newtonschen Parameter beschrieben und die Ergebnisse mit denen für newtonsche Flüssigkeiten verglichen.

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With 9 figures     

Reduced and Generalized Stokes Resolvent Equations in Asymptotically Flat Layers, Part II: H ∞-Calculus

We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. In this second part, we use pseudodifferential operator techniques to construct a parametrix to the reduced Stokes equations, which solves the system in L^q-Sobolev spaces, 1 < q < , modulo terms which get arbitrary small for large resolvent parameters . This parametrix can be analyzed to prove the existence of a bounded H-calculus of the (reduced) Stokes operator.